Back in Time
How many times a day does a 24 hour digital clock look the same when reflected in a horizontal line?
Problem
Beatrix has a 24-hour digital clock on a glass table-top next to her desk.
When she looked at the clock at 13:08, she noticed that the reflected display also read 13:08, as shown.
How many times in a 24-hour period do the display and its reflection give the same time?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
The only digits which will appear the same when reflected in the glass table-top are $0, 1, 3,$ and $8$. So it it necessary to find the number of times in a $24$-hour period that the display on the clock is made up only of these digits.
There are two possibilities for the first digit: $0$ or $1$.
There are four possibilities for the second digit: $0$, $1$, $3$ or $8$.
There are three possibilities for the third digit: $0$, $1$ or $3$.
There are four possibilities for the fourth digit: $0$, $1$, $3$ or $8$.
To find the total number of possible times, we can multiply together the number of possibilities for each digit.
Therefore the display and its reflection give the same time on
$2 \times 4 \times 3 \times 4 =96$ occasions